If you have an ATX power supply, one line (usually green) is the power-on signal. Connecting that line to GND tells the power supply to turn on. There's usually a secondary connection to GND controlled by the motherboard so the supply can turn off automatically when the OS is done with its shutdown procedure.

Check the hardware documentation for your own computer to see how the power control works though.

The LEDs built into the switch all share a common connection to positive voltage. You can connect that to one of the computer's 5V lines. Each LED has its own connection to GND, and will light up when you close that connection.

If you only want to display red, green, or blue, you can make the connections directly to GND. LEDs of different colors operate at different voltages though.. around 1.7V for red LEDs, 2.1V for green LEDs, and 3V for blue LEDs. If you connect the red LED directly to GND, the blue and green LEDs can't light up. The green LED will choke out the blue one, but not the red one.

If you want to mix colors, you can add resistors between the LEDs and GND so each one can operate at its preferred voltage. A 150-Ohm resistor between the green LED and GND, and a 220-Ohm resistor between the red LED and GND should work.